(*
  DaoMath Duality Theory
  对偶性的形式化理论
*)

Require Import Reals.
Require Import InnateBalance.
Open Scope R_scope.

(** * 对偶对的定义 *)

Record DualPair (A : Type) : Type := mkDualPair {
  forward : A;    (* 正向 *)
  backward : A;   (* 反向（对偶） *)
}.

Arguments mkDualPair {A}.
Arguments forward {A}.
Arguments backward {A}.

(** * 对偶性类型类 *)

Class Dualizable (A : Type) := {
  dual : A -> A;
  dual_involutive : forall x, dual (dual x) = x;
}.

(** 实数的对偶性：取负 *)
Instance Real_Dualizable : Dualizable R := {
  dual := fun x => -x;
}.
Proof.
  intros x. lra.
Defined.

(** * 对偶变换性质 *)

(** 定理: 对偶保持零元 *)
Theorem dual_preserves_zero `{Dualizable R} :
  dual 0 = 0.
Proof.
  unfold dual. simpl. lra.
Qed.

(** 定理: 对偶反转加法 *)
Theorem dual_reverses_addition :
  forall x y : R,
  dual (x + y) = dual x + dual y.
Proof.
  intros x y.
  unfold dual. simpl. lra.
Qed.

(** 定理: 对偶反转乘法（需要调整符号） *)
Theorem dual_reverses_multiplication :
  forall x y : R,
  dual (x * y) = (dual x) * (dual y).
Proof.
  intros x y.
  unfold dual. simpl. 
  ring.
Qed.

(** * 对偶对的性质 *)

(** 对偶对的平衡态 *)
Definition balanced_pair (p : DualPair R) : Prop :=
  forward p + backward p = 0.

(** 从先天均衡构造对偶对 *)
Definition pair_from_balance (b : InnateBalance) : DualPair R :=
  mkDualPair (positive b) (negative b).

(** 定理: 从均衡构造的对偶对是平衡的 *)
Theorem pair_from_balance_is_balanced :
  forall b : InnateBalance,
  positive b + negative b = - neutral b ->
  balanced_pair (pair_from_balance b).
Proof.
  intros b H.
  unfold balanced_pair, pair_from_balance. simpl.
  assert (H_sum := innate_sum_zero b).
  lra.
Qed.

(** * 对偶函子 *)

(** 对偶映射 *)
Definition dual_map {A B : Type} 
  `{Dualizable A} `{Dualizable B}
  (f : A -> B) : A -> B :=
  fun x => dual (f (dual x)).

(** 定理: 对偶映射保持恒等 *)
Theorem dual_map_id {A : Type} `{Dualizable A} :
  forall x, dual_map (fun y => y) x = x.
Proof.
  intros x.
  unfold dual_map.
  rewrite dual_involutive.
  rewrite dual_involutive.
  reflexivity.
Qed.

(** * 对偶空间 *)

(** 线性泛函（简化定义） *)
Definition LinearFunctional := R -> R.

(** 对偶空间：线性泛函空间 *)
Definition DualSpace := LinearFunctional.

(** Riesz 表示（简化） *)
Parameter riesz_representation : DualSpace -> R.

(** Riesz 公理 *)
Axiom riesz_axiom : forall (f : DualSpace) (x : R),
  f x = x * (riesz_representation f).

